PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2012), 107–117
DOI: 10.2298/PIM1614107M
SOME CHARACTERIZATION BASED EXPONENTIALITY TESTS AND THEIR BAHADUR EFFICIENCIES
Bojana Milošević and Marko Obradović Abstract. We propose new exponentiality tests based on a recent characterization. We construct integral and Kolmogorov-type statistics, derive their asymptotics and calculate the Bahadur efficiency against some common alternatives. We also obtain a class of locally optimal alternatives for each test. In case of small samples tests are compared with some common exponentiality tests.
1. Introduction Exponential distribution is one of the most exploited distributions thanks to its numerous applications in queueing theory, reliability theory, survival analysis etc. Due to its importance on one hand, and to its numerous suitable properties on the other, the exponential distribution probably has the largest number of characterizing theorems. Many books and chapters are devoted to this topic, e.g., [2, 3, 7, 8]. One of the main directions in goodness-of-fit testing in recent times have become tests based on characterizations. Such tests for exponential distribution are studied in papers [1, 4, 12, 16, 17], among others. In particular, the Bahadur efficiency of such tests has been considered in, e.g., [14, 21, 26, 28]. The characterization we present here is the special case of the characterization from [19]. Let X0 , X1 , X2 be independent and identically distributed non-negative random variables from the distribution whose density f (x) has the Maclaurin expansion for x > 0. If (1.1)
d
X0 + min{X1 , X2 } = max{X1 , X2 }
then f (x) = λe−λx for some λ > 0. 2010 Mathematics Subject Classification: Primary 62G10; 62G20; 62G30. Key words and phrases: goodness-of-fit tests, order statistics, Bahadur efficiency, U -statistics. Supported by Ministry of Education, Science and Technological Development of Republic of Serbia, project 174012. Communicated by Stevan Pilipović. 107
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Let X1 , X2 , . . . , Xn be a random sample from non-negative continuous distribution F . We test the composite hypothesis that F belongs to a family of exponential distributions E(λ), where λ > 0 is an unknown parameter. We consider two test statistics, namely integral-type and Kolmogorov-type. Both of our statistics are invariant with respect to the scale parameter λ (see [15]). Following (1.1) we define two so-called V -empirical distribution functions: Gn (t) =
n n n 1 XXX I{Xi + min(Xj , Xk ) < t}, n3 i=1 j=1 k=1
Hn (t) =
1 n2
n X n X
I{max(Xj , Xk ) < t}.
j=1 k=1
Our test statistics can now be defined as Z ∞ In = (Hn (t) − Gn (t)) dFn (t), 0 Kn = sup Hn (t) − Gn (t) . t>0
We consider large values of our statistics to be significant. The rest of the paper is organized as follows. In Section 2 we examine the asymptotics of integral-type statistic and find Bahadur efficiencies for a choice of common alternatives. In Section 3 we do the analogous study for the Kolomogorovtype statistic. Some classes of locally optimal alternatives are determined in Section 4 and a power study is conducted in Section 5. 2. Integral-type statistic In The statistic In is asymptotically equivalent to U -statistic with symmetric kernel [15] 1 X Ψ(X1 , X2 , X3 , X4 ) = I{max(Xi2 , Xi3 ) < Xi4 } 4! π(4) −I{Xi1 + min(Xi2 , Xi3 ) < Xi4 } , where π(m) is the set of all permutations {i1 , i2 , . . . , im } of set {1, 2, . . . , m}. Its projection on X1 under null hypothesis is
ψ(s) = E(Ψ(X1 , X2 , X3 , X4 )|X1 = s) 1 = P {max(X2 , X3 ) < X4 } − P {s + min(X2 , X3 ) < X4 } 4 1 + P {max(s, X3 ) < X4 } − P {X2 + min(s, X3 ) < X4 } 2 1 + P {max(X3 , X4 ) < s} − P {X2 + min(X3 , X4 ) < s} . 4 After some calculations we get 1 1 1 ψ(s) = − e−2s − e−s . 12 8 6
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The expected value of this projection is equal to zero, while its variance is 1 σI2 = E(ψ 2 (X1 )) = . 1080 Hence this kernel is non-degenerate. Applying Hoeffding’s theorem (see [11]) we √ 2 ). get that the asymptotic distribution of nIn is normal N (0, 135 2.1. Local Bahadur efficiency. The asymptotic efficiency is an established way of measuring the quality of the tests. The Bahadur efficiency has an advantage that it can also be applied to non-normal test statistics, unlike e.g. its Pitman counterpart. For asymptotically normal test statistics these efficiencies coincide (see [6]). The Bahadur efficiency can be expressed as the ratio of the Bahadur exact slope, function describing the rate of exponential decrease for the attained level under the alternative, and double Kullback–Leibler distance between null and alternative distribution. More details on the Bahadur theory can be found in [5,20]. The Bahadur exact slopes are defined as follows. Suppose that the sequence {Tn } of test statistics under alternative converges in probability to some finite function b(θ). Suppose also that the following large deviations limit (2.1)
lim n−1 ln PH0 (Tn > t) = −f (t)
n→∞
exists for any t in an open interval I, on which f is continuous and {b(θ), θ > 0} ⊂ I. Then the Bahadur exact slope is (2.2)
cT (θ) = 2f (b(θ)).
The exact slopes always satisfy the inequality (2.3)
cT (θ) 6 2K(θ),
θ > 0,
where K(θ) is the Kullback–Leibler “distance” between the alternative H1 and the null hypothesis H0 . In view of (2.3), the local Bahadur efficiency of the sequence of statistics Tn is naturally defined as cT (θ) (2.4) eB (T ) = lim . θ→0 2K(θ) The local Bahadur efficiency is measured for alternative distributions that are “close” to the null. Therefore we define the following class of alternatives that are close to exponential. Let G(·, θ), θ > 0, be a family of distributions with densities g(·, θ), such that G(·, 0) is exponential, and the regularity conditions from [20, Chapter 6], and [23, assumptions ND] hold. R∞ Denote h(x) = gθ′ (x, 0). It is obvious that 0 h(x) dx = 0. We now calculate the Bahadur exact slope for the test statistic In . The functions necessary for its calculations are obtained from the following lemmas. Lemma 2.1. For statistic In the function fI from (2.1) is analytic for sufficiently small ε > 0 and it holds 135 2 ε + o(ε2 ), ε → 0. fI (ε) = 2
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Proof. The kernel Ψ is bounded, centered and non-degenerate. Therefore we can apply the theorem of large deviations for non-degenerate U -statistics (see [24]) and get the statement of the lemma. Lemma 2.2. For a given alternative density g(x; θ) whose distribution belongs to G, it holds Z ∞ b(θ) = 4θ ψ(x)h(x) dx + o(θ), θ → 0. 0
Proof. The proof follows from the general result from [23].
The double Kullback–Leibler distance from the densities of the class G to the class of exponential distributions can be for small θ expressed as (see [25]): Z ∞ Z ∞ 2 (2.5) 2K(θ) = h2 (x)ex dx − xh(x) dx · θ2 + o(θ2 ). 0
0
We are going to calculate the local Bahadur efficiency of our test for some common close alternatives. They are: • a Makeham distribution with the density (2.6) g(x; θ) = (1 + θ(1 − e−x )) exp(−x − θ(e−x − 1 + x)), • a Weibull distribution with the density
(2.7)
g(x; θ) = e−x
1+θ
(1 + θ)xθ ,
θ ∈ (0, 1),
θ ∈ (0, 1),
x > 0;
x > 0;
• a gamma distribution with the density (2.8)
(2.9)
(2.10)
g(x; θ) =
xθ e−x , Γ(θ + 1)
θ ∈ (0, 1),
x > 0;
• an exponential mixture with negative weights (EMNW(β)) [13] with density 1 i g(x; θ) = (1 + θ)e−x − βθeβx , θ ∈ 0, , x > 0; β−1 • an exponential distribution with resilience parameter (see [18]) with density g(x; θ) = e−x (1 − e−x )θ (1 + θ),
θ ∈ (0, 1),
x > 0.
In the following example we present calculation of the local Bahadur efficiency. Example 2.1. Let the alternative hypothesis be a Makeham distribution with density function (2.6). The first derivative along θ of its density at θ = 0 is h(x) = −2e−2x + 2e−x − e−x x.
Using (2.5) we get that the Kullback–Leibler distance is K(θ) = θ → 0. Applying Lemma 2.2 we have Z ∞ bI (θ) = 4θ ψ(x)(−2e−2x + 2e−x − e−x x) dx + o(θ) 0
1 = θ + o(θ) ≈ 0.0278θ + o(θ), 36
θ → 0.
1 2 12 θ
+ o(θ2 ),
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According to Lemma 2.1 and (2.4) we get that local Bahadur efficiency eB (I) = 0.625. The calculation procedure for alternatives (2.7)–(2.10) is similar. Therefore we omit it here and present the efficiencies in Table 1. 3. Kolmogorov-type statistic Kn For a fixed t > 0 the expression Hn (t)−Gn (t) is a V-statistic with the following kernel: 1 X I{max(Xi2 , Xi3 ) < t} Ξ(X1 , X2 , X3 , t) = 2! π(3) − I{Xi1 + min(Xi2 , Xi3 ) < t} .
The projection of this family of kernels on X1 under H0 is
ξ(s, t) = E(Ξ(X1 , X2 , X3 , t)|X1 = s) 1 = P {max(X2 , X3 ) < t} − P {s + min(X2 , X3 ) < t} 3 2 + P {max(s, X3 ) < t} − P {X2 + min(s, X3 ) < t} . 3 After some calculations we get 1 1 2 1 ξ(s, t) = e−2t − + te−t + I{s < t}(e−2t+2s + 1 − 2e−t (1 − s + t)). 3 3 3 3 2 The variances of these projections σK (t) under H0 are 4 −4t 2 σK (t) = e (−1 + et )3 . 27 The plot of this function is shown in Figure 1.
2 Figure 1. Plot of the function σK (t),
The supremum is reached for t0 = 1.387, hence 2 2 σK = sup σK (t) = 0.0156. t>0
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Therefore, our family of kernels Ξ(X1 , X2 , X3 , t) is non-degenerate as defined in [22]. Using argumentation of Silverman (see [27]) one can show that U -empirical √ process ρn (t) = n(Hn (t) − Gn (t)), t > 0, weakly converges in D(0, ∞) to a centered Gaussian process ρ(t) with complicated, but calculable covariance. Thus, the sequence of our test statistics Kn converges in distribution to the random variable supt>0 |ρ(t)|. However, we are unable to find its distribution and the critical values should be determined by using Monte Carlo simulations. 3.1. Local Bahadur efficiency. In this subsection we calculate the local Bahadur efficiency for statistic Kn . Similarly to the case of integral-type statistic we determine the large deviation function f and the limit in probability bK (θ) in the following lemmas. Lemma 3.1. For statistic Kn the function fK from (2.1) is analytic for sufficiently small ε > 0 and it holds 1 2 2 2 2 ε → 0. fK (ε) = 2 ε + o(ε ) ≈ 3.56ε + o(ε ), 18σK Proof. The family of kernels {Ξ4 (X1 , X2 , X3 , t), t > 0} is centered and bounded in the sense described in [22]. Applying the large deviation theorem for the supremum of the family of non-degenerate U -and V -statistics (see [22]), we get the statement of the lemma. Lemma 3.2. For a given alternative density g(x; θ) whose distribution belongs to G holds Z ∞ bK (θ) = 3θ sup ξ(x; t)h(x) dx + o(θ), θ → 0. t>0
0
Proof. Using Glivenko–Cantelli theorem for U -statistics (see [9]) we get that the statistic Kn uniformly converges to bK (θ) = sup Pθ {max{X1 , X2 } < t} − Pθ {X1 + min{X2 , X3 } < t} t>0
Z tZ −t 2 = sup (1 − e ) − t>0 0
Denote
a(t; θ) = (1 − e−t )2 −
0
Z tZ 0
t−x
g(x; θ)2(1 − G(y; θ))g(y; θ) dy dx .
t−x
g(x; θ)2(1 − G(y; θ))g(y; θ) dy dx.
0
After some calculations we get that its derivative along θ at θ = 0 is Z ∞ a′θ (t; 0) = 3 ξ(x; t)h(x) dx. 0
Applying Maclaurin’s expansion to the function a(t; θ) we get the statement of the theorem. In the next example we calculate the local Bahadur efficiency in the same manner as we did for integral-type statistic. As before the case of Makeham alternative is presented while for the others the values of efficiencies are given in Table 1.
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Example 3.1. Let the alternative hypothesis be Makeham distribution with density function (2.6). Using Lemma 3.2 we have Z ∞ bK (θ) = sup |a(t, θ)| = 3θ ξ(x, t)(−2e−2x + 2e−x − e−x x) dx + o(θ), θ → 0. 0
t>0
The plot of the function function a′θ (t, 0), is shown in Figure 2.
Figure 2. Plot of the function a′θ (t, 0) Supremum of a(t, θ) is reached at t1 = 1.632, thus bK (θ) = 0.063θ+o(θ), θ → 0. Using Lemma 3.1 and equations (2.2) and (2.4), we get that the local Bahadur efficiency in the case of statistic Kn is 0.342. 4. Locally optimal alternatives The problem of locally optimal alternatives, i.e., the alternatives for which the tests are locally asymptotically optimal in the Bahadur sense, and its importance is described in [20]. In the following theorem we give some classes of such alternatives for our two test statistics. Theorem 4.1. Let g(x; θ) be the density from G that satisfies condition Z ∞ ex h2 (x) dx < ∞. 0
Alternative densities g(x; θ) = e−x + e−x θ(Cψ(x) + D(x − 1)),
x > 0,
C > 0,
D ∈ R,
are for small θ locally asymptotically optimal for the test based on In . Also, alternative densities g(x; θ) = e−x + e−x θ(Cξ(x, t0 ) + D(x − 1)),
x > 0,
C > 0,
D ∈ R,
where t0 = 1.387, are for small θ locally asymptotically optimal for the test based on Kn .
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Proof. Denote h0 (x) = h(x) − (x − 1)e−x
(4.1)
Z
∞
h(s)s ds.
0
It is easy to show that this function satisfies the following equalities. Z ∞ Z ∞ Z ∞ 2 x x 2 2 h0 (x)e dx = e h (x) dx − h(x)x dx ; 0 0 Z0 ∞ Z ∞ ψ(x)h0 (x) dx = ψ(x)h(x) dx; 0 0 Z ∞ Z ∞ ξ(x)h0 (x) dx = ξ(x)h(x) dx. 0
0
The local asymptotic efficiency for the test based on statistic In is 2 2 · 135 cI (θ) 2f (bI (θ)) b2I (θ) 2 bI (θ) = lim = lim = lim θ→0 2K(θ) θ→0 2K(θ) θ→0 θ→0 16σI2 2K(θ) 2K(θ) R 2 ∞ 16θ2 0 ψ(x)h(x) dx + o(θ2 ) = lim R∞ R∞ R∞ 2 θ→0 16 ψ 2 (x)e−x dx θ2 0 ex h2 (x) dx − 0 h(x)x dx + o(θ2 ) 0 2 R∞ 0 ψ(x)h(x) dx = R∞ R∞ R∞ 2 2 −x dx x 2 0 ψ (x)e 0 e h (x) dx − 0 h(x)x dx R∞ 2 ψ(x)h0 (x) dx 0 R∞ = R∞ 2 . ψ (x)e−x dx 0 h20 (x)ex dx 0
eB I = lim
The alternative is locally optimal if eB I = 1. From the Cauchy–Schwarz inequality we get that this holds if and only if h0 (x) = Cψ(x)e−x . Inserting that in (4.1) we obtain h(x). The densities from the statement of the theorem have the same h(x), hence the proof of the first part of the theorem is completed. The local asymptotic efficiency of the test based on statistic Kn is cK (θ) 2f (bK (θ)) b2K (θ) = lim = lim θ→0 2K(θ) θ→0 θ→0 9σK (t0 )2 2K(θ) 2K(θ) R∞ 2 9θ2 supt>0 0 ξ(x, t)h(x) dx + o(θ2 ) = lim R∞ R∞ R∞ 2 θ→0 9 sup 2 −x dx θ 2 x 2 + o(θ2 ) t>0 0 ξ (x, t)e 0 e h (x) dx − 0 h(x)x dx R∞ 2 supt>0 0 ξ(x, t)h(x) dx = R∞ R∞ R∞ 2 supt>0 0 ξ 2 (x, t)e−x dx 0 ex h2 (x) dx − 0 h(x)x dx R∞ 2 supt>0 0 ξ(x, t)h0 (x) dx R∞ = R∞ 2 . ξ (x, t)e−x dx 0 h20 (x)ex dx 0
eK = lim
Once again, using the Cauchy-Schwarz inequality we have that eK = 1 if and only if h0 (x) = Cξ(x, t0 )e−x . Inserting that in (4.1) we obtain h(x). The densities from the statement of the theorem have the same h(x), hence the proof is completed.
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5. Discussion The local Bahadur efficiencies for our test statistics are presented in Table 1. We may notice that the integral type statistic is more efficient than the Kolomogorov type statistic. Comparing with other exponentiality tests based on characterizations (see e.g. [14,28]) we conclude that the In statistic is very efficient and Kn is reasonably efficient (compared to some other Kolmogorov-type tests based on characterizations). Table 1. Local Bahadur efficiency for the statistic In and Kn Alternative Efficiency In Weibull Makeham Gamma EMNW(3) LM GED Resilience
0.750 0.625 0.796 0.844 0.749 0.420 0.803
Efficiency Kn 0.277 0.342 0.161 0.400 0.272 0.247 0.256
6. Power comparison In this section we present empirical powers of our tests for sample sizes n = 20 and n = 50 for some common distributions and compare results with other tests for exponentiality which can be found in [10]. The powers are shown in Tables 2 and 3. The labels used are identical to the ones in [10]. It can be noticed that for sample size n = 20 in majority of the cases the statistic In is the most powerful, and reasonably competitive for n = 50. The powers of statistic Kn are satisfactory. However there are few cases where the powers of both our tests are not suitable. References 1. I. Ahmad, I. Alwasel, A goodness-of-fit test for exponentiality based on the memoryless property, J. R. Stat. Soc., Ser. B, Stat. Methodol. 61(3) (1999), 681–689. 2. M. Ahsanullah, G. G. Hamedani, Exponential Distribution: Theory and Methods, NOVA Science, New York, 2010. 3. B. C. Arnold, N. Balakrishnan, H. N. Nagaraja, A First Course in Order Statistics, SIAM, Philadelphia, 2008. 4. J. E. Angus, Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation, J. Stat. Plann. Inference 6(3) (1982), 241–251. 5. R. R. Bahadur, Some limit theorems in statistics, SIAM, Philadelphia, 1971. 6. , Stochastic comparison of tests, Ann. Math. Stat. 31(2) (1960), 276–295. 7. N. Balakrishnan, C. R. Rao, (eds.), Order Statistics, Theory & Methods, Elsevier, Amsterdam, 1998. 8. J. Galambos, S. Kotz, Characterizations of Probability Distributions, Berlin–Heidelberg–New York, Springer–Verlag, 1978.
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Table 2. Percentage of significant samples for different exponentiality tests n = 20, α = 0.05 Alternative
EP
KS
CM
ω2
KS
KL
S
CO
W(1.4) Γ(2) HN U CH(0.5) CH(1.0) CH(1.5) LF(2.0) LF(4.0) EW(0.5) EW(1.5)
36 48 21 66 63 15 84 28 42 15 45
35 46 24 72 47 18 79 32 44 18 48
35 47 22 70 61 16 83 30 43 16 47
34 47 21 66 61 14 79 28 41 14 43
28 40 18 52 56 13 67 24 34 13 35
29 44 16 61 77 11 76 23 34 11 37
35 46 21 70 63 15 84 29 42 15 46
37 54 19 50 80 13 81 25 37 13 37
I
K
45 35 61 47 24 23 66 81 17 18 28 17 17 18 32 30 47 42 18 17 47 46
Table 3. Percentage of significant samples for different exponentiality tests n = 50, α = 0.05 Alternative
EP
KS
CM
ω2
KS
KL
S
CO
W(1.4) Γ(2) HN U CH(0.5) CH(1.0) CH(1.5) LF(2.0) LF(4.0) EW(0.5) EW(1.5)
80 91 54 98 94 38 100 69 87 38 90
71 86 50 99 90 36 100 65 82 36 88
77 90 53 99 94 37 100 69 87 37 90
75 90 48 98 95 32 100 64 83 32 86
64 83 39 93 92 26 98 53 72 26 75
72 93 37 97 99 23 100 54 75 23 79
79 90 54 99 94 38 100 69 87 38 90
82 96 45 91 99 30 100 60 80 30 78
I
K
83 67 96 83 49 44 96 100 33 32 33 31 32 31 66 58 84 76 33 32 82 82
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(Received 21 01 2016) (Revised 04 07 2016)
